# Differantiation proof question exlanation problem

• transgalactic
In summary, to prove that f(x) is differentiable at x_0, we can use the definition of the derivative: f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}. In this proof, we substitute u = x+h and use the fact that as h approaches 0, u approaches x. This allows us to use the definition of the derivative at the point u instead of x, and we can then simplify the expression to f'(x) = \lim_{h \to 0} \frac{f(x)-f(x-h)}{h}. This is equivalent to the original definition and allows us to use the limit definition with h.
transgalactic
if f(x) is differentiable on x_0
prove that
http://img102.imageshack.us/img102/3189/51290270sj1.th.gif

??

$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

now let $u = x+h \Rightarrow x = u-h$

So $f'(u-h) = \lim_{h \to 0} \frac{f(u-h+h)-f(u-h)}{h} = \lim_{h \to 0} \frac{f(u)-f(u-h)}{h}$

Since $h \to 0, f'(u-h) \to f'(u)$

Therefore $f'(x) = \lim_{h \to 0} \frac{f(x)-f(x-h)}{h}$

$f'(x)+f'(x) = 2f'(x) = \lim_{h \to 0}\left( \frac{f(x+h)-f(x)}{h} + \frac{f(x)-f(x-h)}{h} \right)$

$2f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x-h)}{h}$

Hence, $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x-h)}{2h}$

i understand this solution but
how to pick those variables??
and why it didnt use the point x_0
that they presented

Last edited by a moderator:
there is another definition to limit that uses x_0 that i was given
$f'(x) = \lim_{x \to x_0} \frac{f(x)-f(x_0)}{x-x_0}$

is there a way to use this definition like you did before
what variables to pick so it will go on the "h" defnition??

## 1. What is differentiation?

Differentiation is a mathematical process of finding the rate at which a function changes. It involves calculating the derivative of a function at a specific point, which represents the slope of the tangent line to the function at that point.

## 2. How do you prove a differentiation problem?

To prove a differentiation problem, you need to use the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule, to find the derivative of the given function. Then, you can substitute the values of the independent variable to verify if the derivative is correct.

## 3. What are the common mistakes in differentiation proofs?

Some common mistakes in differentiation proofs include forgetting to apply the chain rule, mixing up the power rule with the product or quotient rule, and making errors in algebraic simplification. It is important to carefully follow the rules of differentiation and double check your work for mistakes.

## 4. How is differentiation used in real life?

Differentiation has many real-life applications, such as in physics, engineering, and economics. It is used to calculate velocities and accelerations, optimize functions to find maximum or minimum values, and analyze rates of change in different scenarios.

## 5. Can differentiation be used to solve optimization problems?

Yes, differentiation can be used to solve optimization problems by finding the critical points of a function, which are the points where the derivative is equal to zero. These points represent the maximum or minimum values of the function, which can be useful in solving real-life problems.

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